Q118 / 160 · TS EAMCET 2022 (20 Jul Shift 2) · Mathematics

$f\left(x, y, c_1, c_2\right)=0$ is an equation containing two arbitrar constants $c_1$ and $c_2$. If the differential equation havin $f\left(x, y, c_1, c_2\right)=0$ as its general solution is of $k^{\text {th }}$ order, the the differential equation corresponding to $x^k+y^k=c^2$ ( $c$ is an arbitrary constant) is

  • Correct answer
    A
    $\frac{d y}{d x}+\frac{x}{y}=0$
  • B
    $\frac{d y}{d x}+\frac{y}{x}=0$
  • C
    $\frac{d y}{d x}-\frac{x}{y}=0$
  • D
    $\frac{d y}{d x}-\frac{y}{x}=0$

Solution

Since the given equation $\mathrm{x}^{\mathrm{k}}+\mathrm{y}^{\mathrm{k}}=\mathrm{c}^2$ has two arbitrary constants $\mathrm{C}_1, \mathrm{C}_2$ hence $\mathrm{k}=2$ $$ \begin{aligned} & \Rightarrow x^2+y^2=C^2 \\ & 2 x+2 y y^1=0 \\ & \Rightarrow y^1=\frac{-x}{y} \\ & \text { or } \frac{d y}{d x}+\frac{x}{y}=0 \end{aligned} $$

Hence, the correct answer is option: 'A'.

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